Ideal hyperbolic polyhedra and discrete uniformization

Abstract

We provide a constructive, variational proof of Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichm\"uller spaces $\widetilde{\mathcal{T}_{g,n}}$ of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over $\mathcal{T}_{g,n}$, and invariant under the action of the mapping class group.